A place to write my way to understanding about issues related to teaching and learning. (Because of my experience, my focus is on mathematics education.) Please join me as I explore the changing educational landscape.

Wednesday, October 26, 2011

At the recent Mathematical Council of the Alberta Teachers' Association [MCATA] Conference, I facilitated the following workshop: Making Mathematical Thinking Visible – Metacognitive Memoirs. Here is the workshop's description:

Metacognition is the awareness of one’s thinking. Memoir is a genre usually referring to a piece of autobiographical writing focusing on some problematic event. Together they represent a powerful tool for helping learners experience what it means to do mathematics by thinking about and communicating their efforts to others. In this session we explore how creative writing supports creative thinking in mathematics – certainly a road less traveled.

In an effort to connect to their prior learning, participants looked over Figure 1 from Cambourne's article, Toward an educationally relevant theory of literacy learning, and identified what was important, what they could connect to, and what questions they had. Cambourne's Conditions were one of the themes of the keynote and I had promised that we would consider them in more detail during this workshop.

Focus: Metacognitive Memoirs [10 minutes]Participants read the following information which allowed them to concentrate on the important aspects of this style of writing in mathematics.

Objective(s): The learner will develop the ability to monitor their thinking as they engage in mathematical problem solving. [According to the book, How Students Learn: Mathematics in the Classroom (National Research Council 2005), this is an essential principle to learning.]

Time: The first demonstration of the approach usually takes a class period. Shared practice, between the students and teacher, may require another couple of class periods. The goal is that after this initial period of scaffolding learners can use the metacognitive mathematical memoirs on a regular basis throughout the remainder of the school year. During the scaffolding, the mathematical content is still being addressed as it is the topic of the memoirs.

Activity: Have the learners look over a group of problems and identify one that represents a ‘just right’ problem – not too hard and not too soft; this might be a problem set they have worked on previously or it might be the first time they have encountered the problem. The idea behind selecting a ‘just right’ problem is that the learners want to be able to tell an interesting story of successfully solving a problem including struggles they encounter along the way.

The learners are asked to describe how they went about solving the problem and what they were thinking as working on it; this represents the metacognitive aspect of the activity. Because memoirs can include embellishments to make the story more interesting, the learners are encouraged to be creative (yet realistic) in describing the “challenges” they encountered along the way. In this way, they are anticipating problems associated with the content.

Pros: (1) I have found these memoirs support learners’ in both a better understanding of the mathematical content and a clearer picture of the process of problem solving; (2) The learners’ writing also provides insight into their thinking – something I often found lacking in their previous assignments, even when they were required to show their work; and (3) The memoirs have been much more interesting for me to read than a solution set – and I only read one or two instead of an entire set.

Cons: (1) Like any new approach, learners often balk at having to write in math class; (2) Also, if learners don’t have experience with metacognition, it can be initially difficult for them to write about their thinking beyond, “I just know it.” It will take time and modeling by the teacher for the learners to become more comfortable showing their thinking as well as their work; and (3) It takes longer to grade a memoir than correct a traditional assignment.

Assessment: For simplicity sake, I often use these four Cs in evaluating learners’ efforts. Learners must have met the expectations at one level to move onto the next. There are opportunities for revision in order to support success.

D

C

B

A

The work is Clear – you have communicated your work with clarity.

The work is Correct – your efforts demonstrate an understanding of mathematical content and/or processes.

The work is Complete – all required aspects of the project have been addressed. There are no gaps in your thinking.

The work is Creative – you present a unique perspective and show you can extend your thinking.

Lessons Learned: Do not try this alone – ask for support from our writing colleagues.

Activity: Choice [30 minutes]Each workshop participant made a decision regarding how to proceed in order to further develop his or her understanding of Metacognitive Memoirs. Some people chose to look at existing models: High School Student (This is the result of a web search from many years ago. I would appreciate any help identifying the source so I can give credit where credit is due.), Preservice Elementary Teacher, and Preservice Secondary Teacher. Others worked on solving and writing their own Metacognitive Memoir using one of these problems: Which is 19/24? or Sowing Seeds.

Reflection: Author's Chair [10 minutes]I asked one of the participants who had been working on writing his own Metacognitive Memoir to share with the group using an approach that I have adapted from literacy instruction.What did he do?He selected to work on "Which is 19/24?" because it related to struggles he was aware of that some of his current students were experiencing. He explained how he used what he knew about common denominators to narrow down the computations he would ultimately have to do. As it turned out, this approach arrived at a solution without having to do any fraction addition or subtraction.So what did he learn?While he wrote in great bit detail how he went about solving the problem using the common denominators as a sorting mechanism, he had not completely shared his thinking. Upon reflection, he recognized that he had thought about and dismissed using other approaches (compute all the items or use a benchmark of 1 to sort the items) because they would not be as efficient. In order to make his thinking visible, he realized that he needed to include this information in his writing.Now what will he do with his learning?Besides being more aware of his thinking (metacognitive) and sharing it with his learners, he also discussed how he might modify the problem so that his learners would have to do some computation. He acknowledged that he had done some deep, conceptual mathematics, but he wanted the problem to address some content skills as well. The other nice thing about using Metacognitive Memoirs is that you can tailor them to the needs of your learners.

Wednesday, October 19, 2011

So far in this series I have discussed the need to empower learners by getting them to ask and explore their own "Now what?" questions (here), considered possible answers to a messy learner-generated word problem (here), and identified implicit conditions associated with the different answers (here). In this final post of the series, I share my preservice teachers' efforts to extend our understanding of one of the possible answers to this word problem:

In the Community, you get two pets. The Elders pick the pets for each family. There were six choices of pets to have: dog, cat, fish, snake, bird, and hamster. What was the probability of getting a dog and a cat?

Three of the possible responses represent combinatoric approaches that the preservice teachers are already familiar with: combinations, permutations, and the multiplication principle. The 1/21 answer (duplicates are allowed but order does not matter), however, represents a new approach.

Many of the learners decide to explore, "How could I generalize this result?" They quickly realize that the 21 comes from adding the combination, 6 choose 2, with the 6 pairs. Consequently, they hypothesize that the general case would be:

n choose 2 + n

In order to check this rule, they try out some simpler problems. 1 pet results in 1 possible pair. 2 pets result in 3 possible pairs. 3 pets result in 6 possible pairs. 4 pets result in 10 possible pairs. 5 pets result in 15 possible pairs.

While this satisfies many of them, a few embrace the idea of extending the problem and notice that the sequence 1, 3, 6, 10, 15, 21, ... looks familiar. In fact, it can be thought of as:

(n + 1) choose 2

This fascinates them and me. I knew this result going into the lesson but I resisted the urge to explore it further before the lesson. "Why does this work?" was a question I did not have an answer to ahead of time. I wanted to work with them instead of guiding them to the answer. (This is an important instructional approach that I wrote about here.)

We are able to show that the two approaches are equivalent fairly easily.

But this still does not explain why (n + 1) choose 2 works when selecting 2 pets from n animals allowing for duplicates.

Finally we begin listing possibilities which leads to us designing the following table. The first entry would be a pair of dogs.

The "+ 1" is the repeat column in the table. This satisfies the "Why?" question but leads us to consider what would happen if 3 pets were selected from 6 animals - allowing for duplicates. Time is up, however, meaning this will be something we can think about on the drive home. Enjoy!

Wednesday, October 12, 2011

Thus far we have considered ways middle school learners can extend their learning by generating their own problems based on young adult literature (here) and how preservice teachers can extend their understanding by considering alternative solutions (here). Given the four different answers they usually come up with (1/15, 1/30, 1/21, and 1/36), the preservice teachers attempt to revise the original problem to match each answer.

For 1/15, the clearer question might be:

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family without any duplication (e.g. no cat-cat pairs).What is the probability of getting a dog and a cat if the order doesn't matter (i.e. cat-dog is the same as dog-cat)?

Typically, this is how the preservice teachers read the original problem even though the original lacks many of the specifics. They see it as a combination problem and so add the necessary conditions in their head. It helps these future teachers to be aware of the implicit conditions hiding in many problems.

The next answer that my learners usually address is 1/30. The clarified question might read:

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family without any duplication (e.g. no cat-cat pairs).What is the probability of getting a dog first and then a cat?

Because the preservice teachers are familiar with permutations this revision is fairly simple for them.

1/36 is usually the third answer they choose work to find the question for:

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family. Duplications, cat-cat pairs, are possible.What is the probability of getting a dog first and then a cat?

This, too, comes quickly since they are comfortable with the multiplication principle.

Finally they address 1/21:

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family. Duplication, cat-cat pairs, are possible.What is the probability of getting a dog and a cat if the order doesn't matter (i.e. cat-dog is the same as dog-cat)?

While they come to this version of the problem easily based on the prior revisions, this is a new context for them and they are eager to explore it further.

Before we look at the math, I engage the preservice teachers in a discussion of the pedagogical worth of having "messy" problems with many possible interpretations. They are inclined to want to clean up problems before sharing them with students but they recognize this is often based on their own experiences in math class. Fortunately, there are usually some voices that identify how considering different points of view made the problem much richer. This ability to identify underlying conditions and considering the alternative problems supports the "Now what?" stance I hope to foster.

We explore the "Now what?" question regarding how to generalize the "1/21 case" in the next post.

Wednesday, October 5, 2011

In the prior post, I introduced a problem written by a seventh-grader as both an example of what middle school students could do when deciding what comes next and an opportunity for preservice teachers to develop and explore their own "Now what?" questions. This was the student-generated problem:

In the Community, you get two pets. The Elders pick the pets for each family. There were six choices of pets to have: dog, cat, fish, snake, bird, and hamster. What was the probability of getting a dog and a cat?

Typically, the preservice teachers came up with an answer fairly quickly. After all, since it comes after a unit on combinations the solution method seems obvious. Still, I ask them to explore the problem further by using one of the extension questions we collect over the course of the semester.

It is sometimes difficult for the preservice teachers to consider alternative answers, however, because of their own experiences with math problems having a single correct answer and the fact they think this problem is so cut-and-dry. Fortunately, I have examples of alternatives to their expected answer of 1/15 that were identified in previous classes. If no one comes up with these alternative answers in the current class, I offer them as other possibilities we ought to consider. I say, "A group came up with an answer of 1/30. Another was pretty sure that it was 1/21, although they also considered 1/36 after they hear the 1/30 rationale."

The preservice teachers' initial reaction is, "Those answers are wrong." I remind them that as educators we must consider that learners are not wrong but they may have answered a question different than what we expected. (I wrote about this here.) Therefore, the natural "Now what?" question that a teacher can consider is, "What question does this answer?"

About Me

I am a professor in the Mathematics Department at Grand Valley State University. Mostly, I teach future teachers but I also do some professional development with inservice middle school teachers. My six-word teaching philosophy is: "Agency and capacity fostering sustainable learning."
My wife, Kathy, is a first grade teacher. She is the person who keeps me grounded in educational reality when I begin to get too idealistic. I have also learned a great deal from her about comprehension strategies and instructional coaching.
I have three adult step-children (Hilary, John, and Andrew).